I was wondering if it is known whether a Vandermonde matrix over a sufficiently large finite field is in general position with respect to intersections of subspaces spanned by subsets of columns, i.e. whether the dimension of such intersections is as small as that of a randomly chosen matrix over a large field. To be more specific, consider a k x n matrix V. Consider any m subsets $S_1$,..., $S_m$ of the set of column indexes {1, 2,... n}. Let $W_i$ be the subspace spanned by the subset of columns with indexes in $S_i$. Let U be the intersection of the subspaces $W_1$,…,$W_m$. The question is whether the dimension of U when matrix V is restricted to be Vandermonde is as small as that when the entries of matrix V are randomly chosen over a large field.

I would be most grateful for any comments, suggestions or pointers to relevant work regarding this problem.